- Email: [email protected]
Heat transfer in a channel at supercritical pressure
WI7
In, .I //curMu\\ Trm/er. Vol.24.No. IO.pp. I661 1674.1981 Printed in Great Br~tam
9310/81:10166708 SO?.l!O/O Pergamon Press Ltd.
HEAT TRANSFER IN A CHANCEL AT SUPERCRITICAL PRESSURE MARIUSZ ADELT*
and JAROSLAW MIKIELEWICZ?
Polish Academy of Sciences, Institute of Fluid Flow Machines, ul. Gen. J. Fiszera 14, R Box 621, W-952
Gdahsk, Poland (Received 14 March 1980 anrf in rer%~rfform 15 Mar& 1981) Abstract-The paper concerns a theoretical analysis ofthe convective heat transfer at supercritical pressures in a channel. The analysis is based on the flow division into two zones with averaged properties the interface between them being the surface of the pseudocritical temperature. Hence, the conservation equations of momentum and energy may be solved separately resulting in analytic formulas for the velocity and temperature profiles and the Nusselt number. The theoretical results are plotted against own experiments with CO, revealing a fairly good agreement.
NOMENCLATURE specific heat at constant
INTRODUCTION
pressure ;
HEAT TRANSFERat
supercritical pressures has become a subject of growing interest in the last 20 years along with the rapid development of energy, rocket and cryogenic technology. Many experimental and theoretical investigations made in this period have been concentrated on the specific features ofsupercritical pressure heat transfer, i.e. crises, local augmentation of heat transfer and oscillation of pressure and flow rate under certain conditions. Essentially the problem of convective heat transfer at supercritical pressure concerns the determination of velocity and temperature profiles in the fluid of variable, strongly temperature dependent properties characteristic for the fluid in the near criticat state (Fig. I). This fact must inevitably be taken into consideration in studies on this heat transfer problem. Strongest variations of all the thermophysical properties occur near the pseudocritical temperature T,, which for a given fluid is a function of pressure alone (similarly as the saturation temperature at subcritical pressures). Specific heat and thermal conductivity reach their maxima at this temperature and density and viscosity reveal maximal variations.
inside tube diameter; gravitational acceleration; specific enthalpy ; thermal conductivity coefficient; mass flow rate ; pressure ; heat flux; radial coordinate (from the axis); axial velocity ; radial coordinate (from the wall); axial coordinate ; temperature; dimensionless radial coordinate ; dimensionless thickness of the laminar sublayer ; turbulent diffusivity ; dynamic viscosity coefficient ; kinematic viscosity coefficient ; density ; shear stress. Subscripts e. h, in, 1, PC. 49 t. W, 7, Superscripts +
experiment ; heavy zone ; inlet ; laminar flow; light zone; pseudo-critical state; heat transfer ; turbulent flow ; theory ; wall ; momentum transfer.
FORMULATION OF THE PROBLEM
The analysis is performed for the supercritical pressure upward flow in a vertical, uniformly heated, circular tube. Turbulent, axially symmetrical, fully developed and steady flow with negligible axial heat conduction and viscous dissipation is assumed. The effect of pressure drop on the fluid properties is also neglected. Under above assumptions the equations of momentum and energy have a form, respectively :
dimensionless ; mean value.
au
PUa,=:-z
* Assistant Professor, Dept. ofThermodynamics and Heat Transfer. + Professor, Dept. OfTherm~ynamics and Heat Transfer. 1667
ap- i$(rT) - pg, ai I3 P@& = ;$‘4)
1668
MARIUSZ ADELT and J*ROSLAW MIKIELEWKZ
FIG.
1. Thermophysical
and the appropriate
properties of CO, in the near-critical region (p from [l].
boundary conditions are:
r=O
?=o
r=R
T=T w
r=O
y=o
r=R
4 = 4,
c?u/ar = 0,
(la)
u = 0,
(lb)
dT/ifr = 0,
CW
T = T,.
@I
Dimensionless velocity and temperature profiles of a turbulent flow are coupled with the shear stress, heat flux and turbulent diffusivity distributions along the channel radius in a manner given by fr’ I I, (3)
dy+, where
(4)
= 7.5
MPa, p/p,
= I .02 I. Data
~termination of the velocity and tem~rature profiles in a channel and, in turn, the shear stress and temperature at the wall requires the solution of the set of non-linear differential equations (l-2) with the turbulent transport parameters i:: and Pr, defined in a certain way. To date, this problem has not been solved analytically, and in important and quite extensive literature one can find a number of numerical solutions with different levels of simplification. Following the relatively comprehensive and critical review of references contained in [ 153 one can point out some characteristic features of previous approaches to the problem under consideration. In the early papers [2-41 velocity and temperature profiles were calculated without referring to the conservation equa:ions. The calculations were based on the assumed heat flux density and shear stress distributions across the channel, e.g. r/s, = 1, r/r, = r/R. The turbulent transport mechanism was described by the formulae originally deduced for constant property fluids and, in different manners, adapted to variable properties. Very simple but descriptive analyses of the shear
Heat
1669
transfer in a channel at supercritical pressure
stress distribution in a variable property flow are presented in [5, 61, where a possibility of the shear stress reduction to zero near the wall due to buoyancy forces or an injection effect is shown, respectively. A long series of investigations involving a fundamental approach based on the conservation equations started in the early 1960’s [7, 81 and is continued along with the progress in numerical methods and electronic computers. These investigations reveal natural trends towards the solution of the conservation equations in their still more complex form and towards better understanding the turbulent transport in variable property fluids. Some recent papers are here a good example. In [9] the authors compared nine different semiempirical models (or their versions) of turbulence. The models were used in the numerical solution ofthe set of the conservation equations written in the form allowing for mixed convection, compressibility and viscous dissipation. The predictions were plotted against experimental data in this way exhibiting the applicability of different models. This comparison may be of help in selecting appropriate semi-empirical models of turbulence when a variable property case is considered. Another approach to the description of turbulence and its relation to variable transport properties, is presented in [lo]. The turbulent diffusivity of momentum and the turbulent Prandtl number in variable property fluids are expressed in terms ofthe differential equations of turbulent kinetic energy and enthalpy pulsations, respectively. However, the equations contain some experimental parameters originally determined for constant property fluids. Using the above model of turbulence the authors solved 2-D conservation equations with the gravity term in the momentum equation and obtained a good agreement with the experimental data for water and air. The same procedure was next repeated in [ll], where the calculations for supercritical water and carbon dioxide were performed. The method proved to be fairly accurate in that case also. A simple method of adapting a constant-property semi-empirical momentum diffusivity to the variable property case was presented and applied in [12, 131. The adaptation was based on the shear stress correction factor resulting from the momentum equation. A set of the 2-D conservation equations was solved numerically in the paper. However, the gravity term was omitted thus limiting the validity of the method to forced convection only. A theoretical analysis of combined forced and free convection heat transfer for turbulent flow of supercritical fluids was presented in [14] which was based on the principle of surface renewal. This relatively new approach follows the instantaneous energy and momentum equations associated with the unsteady transport to individual elements of fluid in residence near the wall. Also this approach leads to predictions for the mean transport properties which are consistent with experimental observations, claim the authors.
A common feature of recent theoretical studies on variable property heat transfer is the extensive use of numerical methods. Obviously, they allow the solving of complex sets of partial differential equations but usually employ large computers and much programming work. This paper presents a new simplified model of supercritical pressure heat transfer and the corresponding analytic method which results in comprehensive flow characteristics with the effect ofbuoyancy forces. The details of the method and experiments are given in [15]. THE MODEL
Inspection of Fig. 1 suggests that within the temperatures T, < T, < T, there exist in the channel cross-section practically two zones of significantly different properties. Low density, viscosity and thermal conductivity fluid layer of the temperature within (T,, T,) attaches the wall. This area is further referred to as the “light” zone. In contrast to this layer the flow core is occupied by the fluid of significantly higher density, viscosity and thermal conductivity thus being referred to as the “heavy” zone. Its temperature varies between T, and T,,. The interface between the both zones is an isothermal rotational surface of the pseudo-critical temperature T,. The idea of the flow divided into two zones is shown in Fig. 2. So far as the heavy zone is concerned it is assumed that the total mass flow rate in this zone remains the same as if the constant property turbulent velocity profile was valid there. On the other hand a substantial difference in enthalpy between the two zones suggests that the heat transferred from the wall is entirely accumulated within the light zone. The second assumption allows one to determine the mass flow rate within the light zone similarly to flows with evaporation : m, =
ndqw(z-zin)
i,- i,
(9)
From the mass balance one obtains m,, = m - ml and accordingly the first assumption on the velocity profile in the heavy zone the cross-section areas occupied by the both zones and thus the geometric coordinate of the interface rpC(y,,) may be determined. The concept of two zones of constant average properties in each zone allows to solve equations (1) and (2) independently and substantially simplifies the problem under consideration. Equation (1) is firstly integrated over the channel area A, the result put back into this equation and the pressure gradient dp/dz thus eliminated. As a result the differential equation of the shear stress distribution in the channel is obtained k:(m)
= %
+ g(p-p),
where 4 PA
J
pdA.
0
(10)
1670
MARIUSZ
ADELT
and
JAROSLAW MI~I~L~wI~~
dimensional formulae for the complete velocity profile are yielded: u; for the laminar sublayer, ui for the turbulent core of the light zone and uh: for the turbulent heavy zone. The only remaining unknown shear stress parameter Re* is numerically determined from algebraic equation (22) obtained by equalizing the known mass flow rate in non-dimensional form with the integral of the non-dimensional velocity profile already predicted :
The two-zone model of supercritical pressure flow.
FIG. 2.
Solving equation (10) with the boundary conditions (equations la, b) and non-dimensionalizing the result one gets the shear stress distribution in both zones: 1 + E--Re*_y+’
?I ?t+ 7 - = D(Re*-y+) %v
7; =;
Ih = F(R+
-y+),
w
(12)
RJ(7W/!d ,
=
(13)
“1
(14) Re
!?
=
(15)
VI
a
d-.-_
4w
Re* Re*-y+
r l-
2
I
F=
1
E 4 Re*-y;
D(Re*-y&j
1
___ R’-y&’
4’ +
< 6,
for
y+ > 6,
(18)
i:: = 0,
(19)
i:: = 0.4y+.
(20)
and
The influence of variable properties on the turbulent diffusivity was taken into account after Goldmann [2] by the relevant definition of the non-dimensional coordinate y& : y+ =
s ’
0
J(rwlp) V
dy.
(R~F+C
_
J,+
)
dy+
.
(23)
and subsequently, the temperature profile within the light zone is determined from equation (2) with the boundary conditions equations (2a, b) by putting the velocity profile and integration. Pr, = 1 was assumed in equation (4). The Nusselt number may be conveniently defined with reference to the constant pseudo-critical temperature T,,:
Nu=4,_=_, d Tw-T,k,
2Re*
Tp’c
(24)
is then expressed by
(17)
The turbulent diffusivity coefficient was estimated after Prandtl’s model as a first approximation of a much more complicated phenomenon : for
- y&’ U+
The wall temperature
E = (1 - D Re*) Re*,
q/q,
2Re* y;
(11)
where Re*
The heat flux distribution
(21)
With the two zone model this coordinate is easily calculated. By introducing equations (11, 12) and (19, 20) into equation (3) and integrating three analytic non-
EXPERIMENTAL
VERIFICATION MODEL
OF THE
For the sake of verification of the theoretical model the experiments were performed and, afterwards, the measured longitudinal wall temperature distributions were compared to the predicted ones [after equation (25)]. The layout of the test rig described in detail in [15] is shown in Fig. 3. The rig is basically a natural circulation loop made of the stainless steel tube of constant diameter within the whole loop. The tube is 10.6mm in inside diameter with the wall 2.2mm in thickness. The test section made ofthe same tube is 1 m in heated length between the bus-bars and is directly DC heated. Uniformly distributed along the section and welded outside the wall are 11 Fe-Ko coated electrically insulated thermocouples. The test section is carefully insulated and the heat losses of the evacuated loop had been measured before the main experiments started. The experiments were performed with CO2 within the following range of parameters: p = 7.5-8.0 MPa; q = 27-llOkW/m’; up = 220
1671
Heat transfer in a channel at supercritical pressure
:
1”” FIG. 3. Layout of the test rig.
-480 kg/m%.
The main characteristics measured were : longitudinal wall temperature distribution (thermocouples), pressure (precision Bourdon pressure gauge) and volumetric flow rate (high accuracy Venturi tube flowmeter). An example of wall temperature distribution measured at constant pressure and variable heat flux is shown in Fig. 4. The experimental Nusselt numbers obtained at maximal and minimal parameters controlling the phenomenon were plotted against their predicted values in Fig. 5. Two thirds of the predicted points fell within the error limit of k 20%, and 3/4 of the points fell within +30x. However, the error distribution along the tube length is not uniform, see Fig. 6. The points appearing higher than the errors correspond either to the heat transfer crisis not accounted for by this model (and probably attributed to relaminarization) or to the uncertainty of the thermal conductivity data in the vicinity of the pseudo-critical temperature. Newer data revealing a local maximum of thermal conductivity in the pseudo-critical (region bounded by the dotted line in Fig. 1) allow a reduction in the error to -20%. The comparison between the results predicted after the theoretical model and three experimental formulae speaks in favour of the model. The Dittus-Boelter formula is involved only as a clear
example of inadequacy of the constant property formula to variable property fluids. On the contrary, the Shitsman formula generated especially for supercritical heat transfer gives reasonably good results but still less accurate than the theory. The same holds for the much newer Protopopov formula [16] based on many experimental data and allowing for both the mixed convection and variable property effects. Thickness ofthe laminar sublayer appears to be very important and it is recommended to take 6 = 5 except for the crisis cross-section. Predicted velocity and temperature profiles at a given heat flux are presented in Fig. 7. In consistency with the assumption taken before, the temperature profiles span over the light zone only. The velocity profiles appear as distinct local maxima due to the buoyant forces. SUMMARY
The theoretical model described in the paper allows one to determine the complete characteristics of the supercritical pressure flow with heat transfer under significant influence of the buoyancy forces. The model is based on an order of simplifications thus giving the solution an approximate character. All the characteristics are given in an analytic form except for the shear
MARIUSZ ADELT
1672
and JAROSLAWMIKIELEWI~Z stress on the wall Re* requiring simple numerical solution ofan algebraic equation. In comparison to the
r
other methods quoted in the References this method is with respect to computer very economical time--complete computations of one channel crosssection take only about 1 min on Polish Odra 1204 computer. The property averaging procedure requires the wall tem~rature to be known Qpriori which is a shortcoming of the met hod when applied in designing. Actually, the wall temperature in technological devices is not known at the beginning. Thus, procedure, starting for example
relevant
iterative
from one of the experimental formulae described in the References, would be of use in that case.
REFERENCES 1. N. V. Vargaftik, Handbook of thermophysical properties of gases and liquids pp. 167-210 (in Russian), Nauka,
Moscow (1972). 2. K. Goldmann, Heat transfer to supercritical water and other fluids with temperature dependent properties, Chem. Engng Prog. Symp. Ser. Nucf. Engng, Pt. 1, %I(11), 105-i 13 (1954). 3. R. G. Deissler, Heat transfer and fluid friction for fully developed turbulent flow of air and supercritical water with variable fluid properties, Trans. ASME 76(l), 73-85 (1954).
Frc;. 4. Measured longitudinal wall temperature distribution at p = 8.OMPa.
4. H. L. Hess and H. R. Kunz, A study of forced convection heat transfer to supercritical hydrogen, Trans. ASME, Ser. C, 87(l), 41-48 (1965). 5. W. B. Hail and J. D. Jackson, Laminarisation of a turbulent pipe flow by buoyancy forces, ASME Paper No. 69-HT-55 (1969). 6. G. N. Kruzhilin, Heat transfer from wall to steam flow at su~r~itical pressures, 5th fnt. Heat Tran.$zr Cant, No. FC 4.11, Tokyo. pp. 173-176 (1974).
FIG. 5. Theoretical vs. experimental Nusselt numbers.
Heat transfer in a channel at supercritical pressure
t
I
JO
20
a4
44
34
64
70
80
90
,
144
2/d
FIG. 6. Distribution of the relative Nusselt number error along the tube length for the model and three experimental formulae.
II I4 . 9-
---i---
e7
s5. 4-
FIG. 7. Predicted dimensionless velocity and temperature profiles at high heat flux in selected cross-sections of the tube.
1673
1674
MARIUSZ AUELT and JAROSLAW MIKIELEWICZ
7. B. S. Petukhov and V. N. Popov, Theoretical determination of heat transfer and shear stress of a turbulent flow of noncompressible variable property fluid in a tube (in Russian). Teplofirika Vysokih Temperatur. l(l), 85-101 (1963). 8. Y. Y. Hsu and J. M. Smith, Theeffect ofdensity variation on heat transfer in the critical region, Tram. ASME, Ser. C 83(2), 176- 183 (1961). 9. B. S. Petukhov, V. D. Vilenskiy and N. V. Medvetskaya, Application of semi-empirical models in heat transfer prediction for turbulent flows of one-phase media of near-critical parameters in tubes (in Russian), Teplqhika Vysokih Temperutur 15(3), 554565 (1977). Turbulent Row 10. B. S. Petukhov and N. V. Medvetskaya, with heat transfer in vertical tubes under a strong influence of buoyant forces (in Russian), Teplqfizika Vvsokih Temuerarur 16(4), 778-786 (1978). 11. B: S. Petukhbv and N.‘V: Medvetskaya, Prediction of turbulent flow and heat transfer in heated tubes for onephase media of near-critical parameters (in Russian), Tepbfizika Vysokih Temperatur 17(2), 343-350 (1979). 12. V. N. Popov, V. M. Relayev and E. P. Baluyeva,
TRANSFERT THERMIQUE
Prediction of heat transfer and fluid friction in the turbulent flow of fluids with different dependence of the transport properties on temperature in a circular tube (in Russian), Teplqfizika Vysokih Temperatur 15(6), 122&1229 (1977). 13. V. N. Popov, V. M. Belayev and E. P. Baluyeva, Prediction of heat transfer and fluid friction in the turbulent flow of helium under supercritical pressure, in a
circular tube (in Russian), Teplqfizika Vysokih Temperatur 16(5), 1018-1027 (1978). 14. C. R. Kakarala and L. C. Thomas, A theoretical analysis of turbulent convective heat transfer for supercritical fluids, 5th Int. Heat Transfer Con$, No. FC 1.10, Tokyo. pp. 45-49 (1974). 15. M. Adelt, Heat transfer and hydromechanics of a supercritical pressure flow (in Polish), Ph.D thesis, Inst. of Fluid Flow Machines, Gdansk, Poland (1978). 16. V. S. Protopopov, Generalized correlations for the local heat transfer coefficients in the turbulent flow of water and carbon dioxide under supercritical pressure in uniformly heated circular tubes (in Russian), Tepbfizika Vysokih Temperafur 15(4), 815-821 (1977).
DANS UN CANAL A PRESSION SUPERCRITIQUE
Resum&On itudie theoriquement la convection thermique a des pressions supercritiques dans un canal. L’analyse est baste sur la division de I’tcoulement en deux zones avec des proprietts moyennes, I’interface
entre elles ttant la surface de la temperature pseudo-critique. Les equations de bilan de quantite de mouvement et d’tnergie peuvent etre resolues separement et conduisent a des formules analytiques pour les profils de vitesse, de temperature et pour le nombre de Nusselt. Les rtsultats theoriques sont compares a des experiences avec CO2 et ils montrent un accord satisfaisant.
wAR~~EUBERTRA~UNG IN EINEMKANALBEI UBERKRITISCHEM
DRUCK
Zusammenfassung-Der Bericht befaBt sich mit einer theoretischen Analyse des konvektiven Warmeilbergangs bei iiberkritischen Drucken in einem Kanal. Die analytische Behandlung beruht auf der Unterteilung der Stromung in zwei Zonen mit gemittelten Eigenschaften, deren Grenzfllche bei der pseudokritischen Temperatur liegt. Daher dtlrfen die Bilanzgleichungen fur Impuls und Energie separat gelost werden, woraus analytische Gleichungen fur Geschwindigkeits- und Temperaturprolile sowie die Nusselt-Zahl erhalten werden. Die theoretischen Ergebnisse werden neben eigenen experimentellen Ergebnissen, die mit CO, gewonnen wurden, dargestellt, wobei eine ziemlich gute Ubereinstimmung festgestellt werden kann.
TEIUIOHEI’LHOC
B KAHAJIE IIPM CBEPXKPMTMqECKOM
flABJIEHMM
AHHoTaunn ~~ AaH TeOpeTHHeCKHH aHaItH3 KOHBeKTHBHOrO TeTUTOnepeHOCa npH CBepXKpHTH’TeCKOM FaBneHHH B KaHane. B OCH~B~ aHanu3a nono~eH0 pa3neneHee noToKa Ha nBe 06nacTH c ocpensesHblMH XapaKTepHcTRKaMH. rpaHHua pasnena MeEny KOTOpbIMH BBnBeTCR nOBepXHOCTbI0 nceBnoKpHTHHeCKOti TeMnepaTypbI, BCneLtCTBHe Her0 ypaBHeHHR COXpaHeHHB BMnynbCa H 3HeprHH MOryT pemaTbca pa3nenbHo. B pe3ynbTaTe nonyqeHbt aHanHTHHecKHe $opMynbT n.nB onpenenewin npothsneii CKOpOCTHH TcMnepaTypbl. a TaKEe ‘TACITaHyCCenbTa. CpaBHeHHe TeOpeTHHeCKHXH 3KCnepHMeHTanbHblX LlaHHblX LUTBCO2 ItaeT flOBOnbH0 XOpOmce COBnafleHHe pe3ynbTaTOB.
In, .I //curMu\\ Trm/er. Vol.24.No. IO.pp. I661 1674.1981 Printed in Great Br~tam
9310/81:10166708 SO?.l!O/O Pergamon Press Ltd.
HEAT TRANSFER IN A CHANCEL AT SUPERCRITICAL PRESSURE MARIUSZ ADELT*
and JAROSLAW MIKIELEWICZ?
Polish Academy of Sciences, Institute of Fluid Flow Machines, ul. Gen. J. Fiszera 14, R Box 621, W-952
Gdahsk, Poland (Received 14 March 1980 anrf in rer%~rfform 15 Mar& 1981) Abstract-The paper concerns a theoretical analysis ofthe convective heat transfer at supercritical pressures in a channel. The analysis is based on the flow division into two zones with averaged properties the interface between them being the surface of the pseudocritical temperature. Hence, the conservation equations of momentum and energy may be solved separately resulting in analytic formulas for the velocity and temperature profiles and the Nusselt number. The theoretical results are plotted against own experiments with CO, revealing a fairly good agreement.
NOMENCLATURE specific heat at constant
INTRODUCTION
pressure ;
HEAT TRANSFERat
supercritical pressures has become a subject of growing interest in the last 20 years along with the rapid development of energy, rocket and cryogenic technology. Many experimental and theoretical investigations made in this period have been concentrated on the specific features ofsupercritical pressure heat transfer, i.e. crises, local augmentation of heat transfer and oscillation of pressure and flow rate under certain conditions. Essentially the problem of convective heat transfer at supercritical pressure concerns the determination of velocity and temperature profiles in the fluid of variable, strongly temperature dependent properties characteristic for the fluid in the near criticat state (Fig. I). This fact must inevitably be taken into consideration in studies on this heat transfer problem. Strongest variations of all the thermophysical properties occur near the pseudocritical temperature T,, which for a given fluid is a function of pressure alone (similarly as the saturation temperature at subcritical pressures). Specific heat and thermal conductivity reach their maxima at this temperature and density and viscosity reveal maximal variations.
inside tube diameter; gravitational acceleration; specific enthalpy ; thermal conductivity coefficient; mass flow rate ; pressure ; heat flux; radial coordinate (from the axis); axial velocity ; radial coordinate (from the wall); axial coordinate ; temperature; dimensionless radial coordinate ; dimensionless thickness of the laminar sublayer ; turbulent diffusivity ; dynamic viscosity coefficient ; kinematic viscosity coefficient ; density ; shear stress. Subscripts e. h, in, 1, PC. 49 t. W, 7, Superscripts +
experiment ; heavy zone ; inlet ; laminar flow; light zone; pseudo-critical state; heat transfer ; turbulent flow ; theory ; wall ; momentum transfer.
FORMULATION OF THE PROBLEM
The analysis is performed for the supercritical pressure upward flow in a vertical, uniformly heated, circular tube. Turbulent, axially symmetrical, fully developed and steady flow with negligible axial heat conduction and viscous dissipation is assumed. The effect of pressure drop on the fluid properties is also neglected. Under above assumptions the equations of momentum and energy have a form, respectively :
dimensionless ; mean value.
au
PUa,=:-z
* Assistant Professor, Dept. ofThermodynamics and Heat Transfer. + Professor, Dept. OfTherm~ynamics and Heat Transfer. 1667
ap- i$(rT) - pg, ai I3 P@& = ;$‘4)
1668
MARIUSZ ADELT and J*ROSLAW MIKIELEWKZ
FIG.
1. Thermophysical
and the appropriate
properties of CO, in the near-critical region (p from [l].
boundary conditions are:
r=O
?=o
r=R
T=T w
r=O
y=o
r=R
4 = 4,
c?u/ar = 0,
(la)
u = 0,
(lb)
dT/ifr = 0,
CW
T = T,.
@I
Dimensionless velocity and temperature profiles of a turbulent flow are coupled with the shear stress, heat flux and turbulent diffusivity distributions along the channel radius in a manner given by fr’ I I, (3)
dy+, where
(4)
= 7.5
MPa, p/p,
= I .02 I. Data
~termination of the velocity and tem~rature profiles in a channel and, in turn, the shear stress and temperature at the wall requires the solution of the set of non-linear differential equations (l-2) with the turbulent transport parameters i:: and Pr, defined in a certain way. To date, this problem has not been solved analytically, and in important and quite extensive literature one can find a number of numerical solutions with different levels of simplification. Following the relatively comprehensive and critical review of references contained in [ 153 one can point out some characteristic features of previous approaches to the problem under consideration. In the early papers [2-41 velocity and temperature profiles were calculated without referring to the conservation equa:ions. The calculations were based on the assumed heat flux density and shear stress distributions across the channel, e.g. r/s, = 1, r/r, = r/R. The turbulent transport mechanism was described by the formulae originally deduced for constant property fluids and, in different manners, adapted to variable properties. Very simple but descriptive analyses of the shear
Heat
1669
transfer in a channel at supercritical pressure
stress distribution in a variable property flow are presented in [5, 61, where a possibility of the shear stress reduction to zero near the wall due to buoyancy forces or an injection effect is shown, respectively. A long series of investigations involving a fundamental approach based on the conservation equations started in the early 1960’s [7, 81 and is continued along with the progress in numerical methods and electronic computers. These investigations reveal natural trends towards the solution of the conservation equations in their still more complex form and towards better understanding the turbulent transport in variable property fluids. Some recent papers are here a good example. In [9] the authors compared nine different semiempirical models (or their versions) of turbulence. The models were used in the numerical solution ofthe set of the conservation equations written in the form allowing for mixed convection, compressibility and viscous dissipation. The predictions were plotted against experimental data in this way exhibiting the applicability of different models. This comparison may be of help in selecting appropriate semi-empirical models of turbulence when a variable property case is considered. Another approach to the description of turbulence and its relation to variable transport properties, is presented in [lo]. The turbulent diffusivity of momentum and the turbulent Prandtl number in variable property fluids are expressed in terms ofthe differential equations of turbulent kinetic energy and enthalpy pulsations, respectively. However, the equations contain some experimental parameters originally determined for constant property fluids. Using the above model of turbulence the authors solved 2-D conservation equations with the gravity term in the momentum equation and obtained a good agreement with the experimental data for water and air. The same procedure was next repeated in [ll], where the calculations for supercritical water and carbon dioxide were performed. The method proved to be fairly accurate in that case also. A simple method of adapting a constant-property semi-empirical momentum diffusivity to the variable property case was presented and applied in [12, 131. The adaptation was based on the shear stress correction factor resulting from the momentum equation. A set of the 2-D conservation equations was solved numerically in the paper. However, the gravity term was omitted thus limiting the validity of the method to forced convection only. A theoretical analysis of combined forced and free convection heat transfer for turbulent flow of supercritical fluids was presented in [14] which was based on the principle of surface renewal. This relatively new approach follows the instantaneous energy and momentum equations associated with the unsteady transport to individual elements of fluid in residence near the wall. Also this approach leads to predictions for the mean transport properties which are consistent with experimental observations, claim the authors.
A common feature of recent theoretical studies on variable property heat transfer is the extensive use of numerical methods. Obviously, they allow the solving of complex sets of partial differential equations but usually employ large computers and much programming work. This paper presents a new simplified model of supercritical pressure heat transfer and the corresponding analytic method which results in comprehensive flow characteristics with the effect ofbuoyancy forces. The details of the method and experiments are given in [15]. THE MODEL
Inspection of Fig. 1 suggests that within the temperatures T, < T, < T, there exist in the channel cross-section practically two zones of significantly different properties. Low density, viscosity and thermal conductivity fluid layer of the temperature within (T,, T,) attaches the wall. This area is further referred to as the “light” zone. In contrast to this layer the flow core is occupied by the fluid of significantly higher density, viscosity and thermal conductivity thus being referred to as the “heavy” zone. Its temperature varies between T, and T,,. The interface between the both zones is an isothermal rotational surface of the pseudo-critical temperature T,. The idea of the flow divided into two zones is shown in Fig. 2. So far as the heavy zone is concerned it is assumed that the total mass flow rate in this zone remains the same as if the constant property turbulent velocity profile was valid there. On the other hand a substantial difference in enthalpy between the two zones suggests that the heat transferred from the wall is entirely accumulated within the light zone. The second assumption allows one to determine the mass flow rate within the light zone similarly to flows with evaporation : m, =
ndqw(z-zin)
i,- i,
(9)
From the mass balance one obtains m,, = m - ml and accordingly the first assumption on the velocity profile in the heavy zone the cross-section areas occupied by the both zones and thus the geometric coordinate of the interface rpC(y,,) may be determined. The concept of two zones of constant average properties in each zone allows to solve equations (1) and (2) independently and substantially simplifies the problem under consideration. Equation (1) is firstly integrated over the channel area A, the result put back into this equation and the pressure gradient dp/dz thus eliminated. As a result the differential equation of the shear stress distribution in the channel is obtained k:(m)
= %
+ g(p-p),
where 4 PA
J
pdA.
0
(10)
1670
MARIUSZ
ADELT
and
JAROSLAW MI~I~L~wI~~
dimensional formulae for the complete velocity profile are yielded: u; for the laminar sublayer, ui for the turbulent core of the light zone and uh: for the turbulent heavy zone. The only remaining unknown shear stress parameter Re* is numerically determined from algebraic equation (22) obtained by equalizing the known mass flow rate in non-dimensional form with the integral of the non-dimensional velocity profile already predicted :
The two-zone model of supercritical pressure flow.
FIG. 2.
Solving equation (10) with the boundary conditions (equations la, b) and non-dimensionalizing the result one gets the shear stress distribution in both zones: 1 + E--Re*_y+’
?I ?t+ 7 - = D(Re*-y+) %v
7; =;
Ih = F(R+
-y+),
w
(12)
RJ(7W/!d ,
=
(13)
“1
(14) Re
!?
=
(15)
VI
a
d-.-_
4w
Re* Re*-y+
r l-
2
I
F=
1
E 4 Re*-y;
D(Re*-y&j
1
___ R’-y&’
4’ +
< 6,
for
y+ > 6,
(18)
i:: = 0,
(19)
i:: = 0.4y+.
(20)
and
The influence of variable properties on the turbulent diffusivity was taken into account after Goldmann [2] by the relevant definition of the non-dimensional coordinate y& : y+ =
s ’
0
J(rwlp) V
dy.
(R~F+C
_
J,+
)
dy+
.
(23)
and subsequently, the temperature profile within the light zone is determined from equation (2) with the boundary conditions equations (2a, b) by putting the velocity profile and integration. Pr, = 1 was assumed in equation (4). The Nusselt number may be conveniently defined with reference to the constant pseudo-critical temperature T,,:
Nu=4,_=_, d Tw-T,k,
2Re*
Tp’c
(24)
is then expressed by
(17)
The turbulent diffusivity coefficient was estimated after Prandtl’s model as a first approximation of a much more complicated phenomenon : for
- y&’ U+
The wall temperature
E = (1 - D Re*) Re*,
q/q,
2Re* y;
(11)
where Re*
The heat flux distribution
(21)
With the two zone model this coordinate is easily calculated. By introducing equations (11, 12) and (19, 20) into equation (3) and integrating three analytic non-
EXPERIMENTAL
VERIFICATION MODEL
OF THE
For the sake of verification of the theoretical model the experiments were performed and, afterwards, the measured longitudinal wall temperature distributions were compared to the predicted ones [after equation (25)]. The layout of the test rig described in detail in [15] is shown in Fig. 3. The rig is basically a natural circulation loop made of the stainless steel tube of constant diameter within the whole loop. The tube is 10.6mm in inside diameter with the wall 2.2mm in thickness. The test section made ofthe same tube is 1 m in heated length between the bus-bars and is directly DC heated. Uniformly distributed along the section and welded outside the wall are 11 Fe-Ko coated electrically insulated thermocouples. The test section is carefully insulated and the heat losses of the evacuated loop had been measured before the main experiments started. The experiments were performed with CO2 within the following range of parameters: p = 7.5-8.0 MPa; q = 27-llOkW/m’; up = 220
1671
Heat transfer in a channel at supercritical pressure
:
1”” FIG. 3. Layout of the test rig.
-480 kg/m%.
The main characteristics measured were : longitudinal wall temperature distribution (thermocouples), pressure (precision Bourdon pressure gauge) and volumetric flow rate (high accuracy Venturi tube flowmeter). An example of wall temperature distribution measured at constant pressure and variable heat flux is shown in Fig. 4. The experimental Nusselt numbers obtained at maximal and minimal parameters controlling the phenomenon were plotted against their predicted values in Fig. 5. Two thirds of the predicted points fell within the error limit of k 20%, and 3/4 of the points fell within +30x. However, the error distribution along the tube length is not uniform, see Fig. 6. The points appearing higher than the errors correspond either to the heat transfer crisis not accounted for by this model (and probably attributed to relaminarization) or to the uncertainty of the thermal conductivity data in the vicinity of the pseudo-critical temperature. Newer data revealing a local maximum of thermal conductivity in the pseudo-critical (region bounded by the dotted line in Fig. 1) allow a reduction in the error to -20%. The comparison between the results predicted after the theoretical model and three experimental formulae speaks in favour of the model. The Dittus-Boelter formula is involved only as a clear
example of inadequacy of the constant property formula to variable property fluids. On the contrary, the Shitsman formula generated especially for supercritical heat transfer gives reasonably good results but still less accurate than the theory. The same holds for the much newer Protopopov formula [16] based on many experimental data and allowing for both the mixed convection and variable property effects. Thickness ofthe laminar sublayer appears to be very important and it is recommended to take 6 = 5 except for the crisis cross-section. Predicted velocity and temperature profiles at a given heat flux are presented in Fig. 7. In consistency with the assumption taken before, the temperature profiles span over the light zone only. The velocity profiles appear as distinct local maxima due to the buoyant forces. SUMMARY
The theoretical model described in the paper allows one to determine the complete characteristics of the supercritical pressure flow with heat transfer under significant influence of the buoyancy forces. The model is based on an order of simplifications thus giving the solution an approximate character. All the characteristics are given in an analytic form except for the shear
MARIUSZ ADELT
1672
and JAROSLAWMIKIELEWI~Z stress on the wall Re* requiring simple numerical solution ofan algebraic equation. In comparison to the
r
other methods quoted in the References this method is with respect to computer very economical time--complete computations of one channel crosssection take only about 1 min on Polish Odra 1204 computer. The property averaging procedure requires the wall tem~rature to be known Qpriori which is a shortcoming of the met hod when applied in designing. Actually, the wall temperature in technological devices is not known at the beginning. Thus, procedure, starting for example
relevant
iterative
from one of the experimental formulae described in the References, would be of use in that case.
REFERENCES 1. N. V. Vargaftik, Handbook of thermophysical properties of gases and liquids pp. 167-210 (in Russian), Nauka,
Moscow (1972). 2. K. Goldmann, Heat transfer to supercritical water and other fluids with temperature dependent properties, Chem. Engng Prog. Symp. Ser. Nucf. Engng, Pt. 1, %I(11), 105-i 13 (1954). 3. R. G. Deissler, Heat transfer and fluid friction for fully developed turbulent flow of air and supercritical water with variable fluid properties, Trans. ASME 76(l), 73-85 (1954).
Frc;. 4. Measured longitudinal wall temperature distribution at p = 8.OMPa.
4. H. L. Hess and H. R. Kunz, A study of forced convection heat transfer to supercritical hydrogen, Trans. ASME, Ser. C, 87(l), 41-48 (1965). 5. W. B. Hail and J. D. Jackson, Laminarisation of a turbulent pipe flow by buoyancy forces, ASME Paper No. 69-HT-55 (1969). 6. G. N. Kruzhilin, Heat transfer from wall to steam flow at su~r~itical pressures, 5th fnt. Heat Tran.$zr Cant, No. FC 4.11, Tokyo. pp. 173-176 (1974).
FIG. 5. Theoretical vs. experimental Nusselt numbers.
Heat transfer in a channel at supercritical pressure
t
I
JO
20
a4
44
34
64
70
80
90
,
144
2/d
FIG. 6. Distribution of the relative Nusselt number error along the tube length for the model and three experimental formulae.
II I4 . 9-
---i---
e7
s5. 4-
FIG. 7. Predicted dimensionless velocity and temperature profiles at high heat flux in selected cross-sections of the tube.
1673
1674
MARIUSZ AUELT and JAROSLAW MIKIELEWICZ
7. B. S. Petukhov and V. N. Popov, Theoretical determination of heat transfer and shear stress of a turbulent flow of noncompressible variable property fluid in a tube (in Russian). Teplofirika Vysokih Temperatur. l(l), 85-101 (1963). 8. Y. Y. Hsu and J. M. Smith, Theeffect ofdensity variation on heat transfer in the critical region, Tram. ASME, Ser. C 83(2), 176- 183 (1961). 9. B. S. Petukhov, V. D. Vilenskiy and N. V. Medvetskaya, Application of semi-empirical models in heat transfer prediction for turbulent flows of one-phase media of near-critical parameters in tubes (in Russian), Teplqhika Vysokih Temperutur 15(3), 554565 (1977). Turbulent Row 10. B. S. Petukhov and N. V. Medvetskaya, with heat transfer in vertical tubes under a strong influence of buoyant forces (in Russian), Teplqfizika Vvsokih Temuerarur 16(4), 778-786 (1978). 11. B: S. Petukhbv and N.‘V: Medvetskaya, Prediction of turbulent flow and heat transfer in heated tubes for onephase media of near-critical parameters (in Russian), Tepbfizika Vysokih Temperatur 17(2), 343-350 (1979). 12. V. N. Popov, V. M. Relayev and E. P. Baluyeva,
TRANSFERT THERMIQUE
Prediction of heat transfer and fluid friction in the turbulent flow of fluids with different dependence of the transport properties on temperature in a circular tube (in Russian), Teplqfizika Vysokih Temperatur 15(6), 122&1229 (1977). 13. V. N. Popov, V. M. Belayev and E. P. Baluyeva, Prediction of heat transfer and fluid friction in the turbulent flow of helium under supercritical pressure, in a
circular tube (in Russian), Teplqfizika Vysokih Temperatur 16(5), 1018-1027 (1978). 14. C. R. Kakarala and L. C. Thomas, A theoretical analysis of turbulent convective heat transfer for supercritical fluids, 5th Int. Heat Transfer Con$, No. FC 1.10, Tokyo. pp. 45-49 (1974). 15. M. Adelt, Heat transfer and hydromechanics of a supercritical pressure flow (in Polish), Ph.D thesis, Inst. of Fluid Flow Machines, Gdansk, Poland (1978). 16. V. S. Protopopov, Generalized correlations for the local heat transfer coefficients in the turbulent flow of water and carbon dioxide under supercritical pressure in uniformly heated circular tubes (in Russian), Tepbfizika Vysokih Temperafur 15(4), 815-821 (1977).
DANS UN CANAL A PRESSION SUPERCRITIQUE
Resum&On itudie theoriquement la convection thermique a des pressions supercritiques dans un canal. L’analyse est baste sur la division de I’tcoulement en deux zones avec des proprietts moyennes, I’interface
entre elles ttant la surface de la temperature pseudo-critique. Les equations de bilan de quantite de mouvement et d’tnergie peuvent etre resolues separement et conduisent a des formules analytiques pour les profils de vitesse, de temperature et pour le nombre de Nusselt. Les rtsultats theoriques sont compares a des experiences avec CO2 et ils montrent un accord satisfaisant.
wAR~~EUBERTRA~UNG IN EINEMKANALBEI UBERKRITISCHEM
DRUCK
Zusammenfassung-Der Bericht befaBt sich mit einer theoretischen Analyse des konvektiven Warmeilbergangs bei iiberkritischen Drucken in einem Kanal. Die analytische Behandlung beruht auf der Unterteilung der Stromung in zwei Zonen mit gemittelten Eigenschaften, deren Grenzfllche bei der pseudokritischen Temperatur liegt. Daher dtlrfen die Bilanzgleichungen fur Impuls und Energie separat gelost werden, woraus analytische Gleichungen fur Geschwindigkeits- und Temperaturprolile sowie die Nusselt-Zahl erhalten werden. Die theoretischen Ergebnisse werden neben eigenen experimentellen Ergebnissen, die mit CO, gewonnen wurden, dargestellt, wobei eine ziemlich gute Ubereinstimmung festgestellt werden kann.
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